Finetti's Representation Theorem Research Articles

Finitely additive mixtures of probability measures

Let D be a linear space of real bounded functions and P:D→R a coherent functional. Also, let Q be a collection of coherent functionals on D. Under mild conditions, there is a finitely additive probability Π on the power set of Q such that P(f)=∫QQ(f)Π(dQ) for each f∈D. This fact has various consequences and such consequences are investigated in this paper. Three types of results are provided: (i) Existence of common extensions satisfying certain properties, (ii) Finitely additive mixtures of extreme points, (iii) Countably additive mixtures. Among other things, we obtain new versions of Kolmogorov's consistency theorem and de Finetti's representation theorem.

An information-theoretic proof of a finite de Finetti theorem

A finite form of de Finetti's representation theorem is established using elementary information-theoretic tools: The distribution of the first $k$ random variables in an exchangeable binary vector of length $n\geq k$ is close to a mixture of product distributions. Closeness is measured in terms of the relative entropy and an explicit bound is provided.

Exchangeable choice functions

We investigate how to model exchangeability with choice functions. Exchangeability is a structural assessment on a sequence of uncertain variables. We show how such assessments constitute a special kind of indifference assessment, and how this idea leads to a counterpart of de Finetti's Representation Theorem, both in a finite and a countable context.

On the rate of convergence in de Finetti’s representation theorem

A consequence of de Finetti's representation theorem is that for every infinite sequence of exchangeable 0-1 random variables $(X_k)_{k\geq1}$, there exists a probability measure $\mu$ on the Borel sets of $[0,1]$ such that $\bar X_n = n^{-1} \sum_{i=1}^n X_i$ converges weakly to $\mu$. For a wide class of probability measures $\mu$ having smooth density on $(0,1)$, we give bounds of order $1/n$ with explicit constants for the Wasserstein distance between the law of $\bar X_n$ and $\mu$. This extends a recent result {by} Goldstein and Reinert \cite{goldstein2013stein} regarding the distance between the scaled number of white balls drawn in a P\'olya-Eggenberger urn and its limiting distribution. We prove also that, in the most general cases, the distance between the law of $\bar X_n$ and $\mu$ is bounded below by $1/n$ and above by $1/\sqrt{n}$ (up to some multiplicative constants). For every $\delta \in [1/2,1]$, we give an example of an exchangeable sequence such that this distance is of order $1/n^\delta$.

Coherent Predictive Inference under Exchangeability with Imprecise Probabilities

Coherent reasoning under uncertainty can be represented in a very general manner by coherent sets of desirable gambles. In a context that does not allow for indecision, this leads to an approach that is mathematically equivalent to working with coherent conditional probabilities. If we do allow for indecision, this leads to a more general foundation for coherent (imprecise-)probabilistic inference. In this framework, and for a given finite category set, coherent predictive inference under exchangeability can be represented using Bernstein coherent cones of multivariate polynomials on the simplex generated by this category set. This is a powerful generalisation of de Finetti's Representation Theorem allowing for both imprecision and indecision. We define an inference system as a map that associates a Bernstein coherent cone of polynomials with every finite category set. Many inference principles encountered in the literature can then be interpreted, and represented mathematically, as restrictions on such maps. We discuss, as particular examples, two important inference principles: representation insensitivity—a strengthened version of Walley's representation invariance—and specificity. We show that there is an infinity of inference systems that satisfy these two principles, amongst which we discuss in particular the skeptically cautious inference system, the inference systems corresponding to (a modified version of) Walley and Bernard's Imprecise Dirichlet Multinomial Models (IDMM), the skeptical IDMM inference systems, and the Haldane inference system. We also prove that the latter produces the same posterior inferences as would be obtained using Haldane's improper prior, implying that there is an infinity of proper priors that produce the same coherent posterior inferences as Haldane's improper one. Finally, we impose an additional inference principle that allows us to characterise uniquely the immediate predictions for the IDMM inference systems.

On exchangeability and Hausdorff moment problems over k-dimensional simplexes

Two classical problems, the Hausdorff moment problem and de Finetti's representation theorem for exchangeable random variables, are considered. We generalize these problems to a g-tuple of k-dimensional simplexes and an infinite sequence of g-fold partially exchangeable random variables in {0,1,...,k}, respectively. The equivalence between them is then formalized and proven.

Limit distributions of scale-invariant probabilistic models of correlated random variables with the q-Gaussian as an explicit example

Extremization of the Boltzmann-Gibbs (BG) entropy under appropriate norm and width constraints yields the Gaussian distribution. Also, the basic solutions of the standard Fokker-Planck (FP) equation (related to the Langevin equation with additive noise), as well as the Central Limit Theorem attractors, are Gaussians. The simplest stochastic model with such features is N to infinity independent binary random variables, as first proved by de Moivre and Laplace. What happens for strongly correlated random variables? Such correlations are often present in physical situations as e.g. systems with long range interactions or memory. Frequently q-Gaussians become observed. This is typically so if the Langevin equation includes multiplicative noise, or the FP equation to be nonlinear. Scale-invariance, i.e. exchangeable binary stochastic processes, allow a systematical analysis of the relation between correlations and non-Gaussian distributions. In particular, a generalized stochastic model yielding q-Gaussians for all q (including q>1) was missing. This is achieved here by using the Laplace-de Finetti representation theorem, which embodies strict scale-invariance of interchangeable random variables. We demonstrate that strict scale invariance together with q-Gaussianity mandates the associated extensive entropy to be BG.

De Finetti Representation Theorem for Infinite-Dimensional Quantum Systems and Applications to Quantum Cryptography

We show that the quantum de Finetti theorem holds for states on infinite-dimensional systems, provided they satisfy certain experimentally verifiable conditions. This result can be applied to prove the security of quantum key distribution based on weak coherent states or other continuous variable states against general attacks.

SECURITY OF QUANTUM KEY DISTRIBUTION

Quantum Information Theory is an area of physics which studies both fundamental and applied issues in quantum mechanics from an information-theoretical viewpoint. The underlying techniques are, however, often restricted to the analysis of systems which satisfy a certain independence condition. For example, it is assumed that an experiment can be repeated independently many times or that a large physical system consists of many virtually independent parts. Unfortunately, such assumptions are not always justified. This is particularly the case for practical applications — e.g. in quantum cryptography — where parts of a system might have an arbitrary and unknown behavior. We propose an approach which allows us to study general physical systems for which the above mentioned independence condition does not necessarily hold. It is based on an extension of various information-theoretical notions. For example, we introduce new uncertainty measures, called smooth min- and max-entropy, which are generalizations of the von Neumann entropy. Furthermore, we develop a quantum version of de Finetti's representation theorem, as described below. Consider a physical system consisting of n parts. These might, for instance, be the outcomes of n runs of a physical experiment. Moreover, we assume that the joint state of this n-partite system can be extended to an (n + k)-partite state which is symmetric under permutations of its parts (for some k ≫ 1). The de Finetti representation theorem then says that the original n-partite state is, in a certain sense, close to a mixture of product states. Independence thus follows (approximatively) from a symmetry condition. This symmetry condition can easily be met in many natural situations. For example, it holds for the joint state of n parts, which are chosen at random from an arbitrary (n + k)-partite system. As an application of these techniques, we prove the security of quantum key distribution (QKD), i.e. secret key agreement by communication over a quantum channel. In particular, we show that, in order to analyze QKD protocols, it is generally sufficient to consider so-called collective attacks, where the adversary is restricted to applying the same operation to each particle sent over the quantum channel separately. The proof is generic and thus applies to known protocols such as BB84 and B92 (where better bounds on the secret-key rate and on the the maximum tolerated noise level of the quantum channel are obtained) as well as to continuous variable schemes (where no full security proof has been known). Furthermore, the security holds with respect to a strong so-called universally composable definition. This implies that the keys generated by a QKD protocol can safely be used in any application, e.g. for one-time pad encryption — which, remarkably, is not the case for most standard definitions.

Finite de Finetti Theorem for Infinite-Dimensional Systems

We formulate and prove a de Finetti representation theorem for finitely exchangeable states of a quantum system consisting of k infinite-dimensional subsystems. The theorem is valid for states that can be written as the partial trace of a pure state |Psi/Psi| chosen from a family of subsets {Cn} of the full symmetric subspace for n subsystems. We show that such states become arbitrarily close to mixtures of pure power states as n increases. We give a second equivalent characterization of the family {Cn}.

De Finetti representation theorem for quantum-process tomography

In quantum process tomography, it is possible to express the experimenter's prior information as a sequence of quantum operations, i.e., trace-preserving completely positive maps. In analogy to de Finetti's concept of exchangeability for probability distributions, we give a definition of exchangeability for sequences of quantum operations. We then state and prove a representation theorem for such exchangeable sequences. The theorem leads to a simple characterization of admissible priors for quantum process tomography and solves to a Bayesian's satisfaction the problem of an unknown quantum operation.

A statistical treatment of the problem of division

The problem of division is one of the most important problems in the emergence of probability. It has been long considered “solved” from a probabilistic viewpoint. However, we do not find the solution satisfactory. In this study, the problem is recasted as a statistical problem. The outcomes of matches of the game are considered as an infinitely exchangeable random sequence and predictors/estimators are constructed in light of de Finetti representation theorem. Bounds of the estimators are derived over wide classes of priors (mixing distributions). We find that, although conservative, the classical solutions are justifiable by our analysis while the plug-in estimates are too optimistic for the winning player.

Unknown quantum states: The quantum de Finetti representation

We present an elementary proof of the quantum de Finetti representation theorem, a quantum analog of de Finetti’s classical theorem on exchangeable probability assignments. This contrasts with the original proof of Hudson and Moody [Z. Wahrschein. verw. Geb. 33, 343 (1976)], which relies on advanced mathematics and does not share the same potential for generalization. The classical de Finetti theorem provides an operational definition of the concept of an unknown probability in Bayesian probability theory, where probabilities are taken to be degrees of belief instead of objective states of nature. The quantum de Finetti theorem, in a closely analogous fashion, deals with exchangeable density-operator assignments and provides an operational definition of the concept of an “unknown quantum state” in quantum-state tomography. This result is especially important for information-based interpretations of quantum mechanics, where quantum states, like probabilities, are taken to be states of knowledge rather than states of nature. We further demonstrate that the theorem fails for real Hilbert spaces and discuss the significance of this point.

Some results on the representation of measures of location and spread as L-functionals

Random variables are compared w.r.t. their location or their dispersion by means of various types of stochastic orderings. First degree stochastic dominance is relevant to location and the spread ordering, defined via comparisons of all the corresponding interquantile distances, to dispersion. Our first result is a characterization of some measures of location as L-functionals, and was inspired by de Finetti's representation theorem for quasi-linear means. The main result of the paper is a representation theorem for some spread measures as integrals with respect to the measure defined by the quantile function. A ‘spread distance’, namely a metric linked to the ordering, is introduced and continuity of the functional with respect to this metric is required to be one of the characteristic properties that enable the representation to be made.

Some critical aspects of the use of exchangeability in statistics

This expository paper provides a framework for analysing de Finetti's representation theorem for exchangeable finitely additive probabilities. Such an analysis is justified by reasoning of statistical nature, since it is shown that the abandonment of the axiom of σ-additivity has some noteworthy consequences on the common interpretation of the Bayesian paradigm. The usual (strong) fromulation of de Finetti's theorem is deduced from the finitely additive (weak) formulation, and it is used to solve the problem of stating the existence of a stochastic process, with given finite-dimensional probability distributions, whose sample paths are probability distributions. It is of importance, in particular, to specify prior distributions for nonparametric inferential problems in a Bayesian setting.

Predicting the unpredictable

A major difficulty for currently existing theories of inductive inference involves the question of what to do when novel, unknown, or previously unsuspected phenomena occur. In this paper one particular instance of this difficulty is considered, the so-called sampling of species problem. The classical probabilistic theories of inductive inference due to Laplace, Johnson, de Finetti, and Carnap adopt a model of simple enumerative induction in which there are a prespecified number of types or species which may be observed. But, realistically, this is often not the case. In 1838 the English mathematician Augustus De Morgan proposed a modification of the Laplacian model to accommodate situations where the possible types or species to be observed are not assumed to be known in advance; but he did not advance a justification for his solution. In this paper a general philosophical approach to such problems is suggested, drawing on work of the English mathematician J. F. C. Kingman. It then emerges that the solution advanced by De Morgan has a very deep, if not totally unexpected, justification. The key idea is that although ‘exchangeable’ random sequences are the right objects to consider when all possible outcome-types are known in advance, exchangeable random partitions are the right objects to consider when they are not. The result turns out to be very satisfying. The classical theory has several basic elements: a representation theorem for the general exchangeable sequence (the de Finetti representation theorem), a distinguished class of sequences (those employing Dirichlet priors), and a corresponding rule of succession (the continuum of inductive methods). The new theory has parallel basic elements: a representation theorem for the general exchangeable random partition (the Kingman representation theorem), a distinguished class of random partitions (the Poisson-Dirichlet process), and a rule of succession which corresponds to De Morgan's rule.

The significance of the ergodic decomposition of stationary measures for the interpretation of probability

De Finetti's representation theorem is a special case of the ergodic decomposition of stationary probability measures. The problems of the interpretation of probabilities centred around de Finetti's theorem are extended to this more general situation. The ergodic decomposition theorem has a physical background in the ergodic theory of dynamical systems. Thereby the interpretations of probabilities in the cases of de Finetti's theorem and its generalization and in ergodic theory are systematically connected to each other.

The Generalization of de Finetti's Representation Theorem to Stationary Probabilities

According to de Finetti's representation theorem, probabilities of exchangeable sequences of events are unique mixtures of Bernoullian probabilities. The exchangeable probability P(E) of the sequence E equals an integral of the form , where Px(E) is for each x (0≦x≦1) a Bernoullian measure with the probability x for ‘success’ and independence between consecutive events, and F a uniquely determined distribution of the parameter x. P(E) equals the expected value of Px(E). If F has a derivative f, Probabilities can be calculated under the integral sign as if events were independent and had a constant success probability. Therefore there cannot be anything one can do with Bernoullian probabilities, but could not do with the exchangeable ones, de Finetti thinks that he has, because of the unique correspondence between exchangeable measures and distributions over a parameter each value of which gives a Bernoullian probability, shown the unnecessity of the latter.

Unknown Probabilities, Bayesianism, and de Finetti's Representation Theorem

In the development of the theory of subjective probability and in discussions about it an important role has been played by de Finetti's representation theorem and by a number of related results presented in de Finetti's classic monograph. This theorem, together with the notion of exchangeability (among its aliases there are the terms equivalence, permutability, and symmetry), was originally put forward by Bruno de Finetti as a solution to a problem to which his subjectivistic interpretation of probability had led him. Although this theorem is mentioned in many expositions of the theory of subjective probability, there are not many satisfactory discussions of its significance available to philosophers. (The best one, and almost the only one, is Braithwaite's paper ‘On Unknown Probabilities’.)